Demystifying Orthogonal Monte Carlo and Beyond
arXiv:2005.13590
Abstract
Orthogonal Monte Carlo (OMC) is a very effective sampling algorithm imposing structural geometric conditions (orthogonality) on samples for variance reduction. Due to its simplicity and superior performance as compared to its Quasi Monte Carlo counterparts, OMC is used in a wide spectrum of challenging machine learning applications ranging from scalable kernel methods to predictive recurrent neural networks, generative models and reinforcement learning. However theoretical understanding of the method remains very limited. In this paper we shed new light on the theoretical principles behind OMC, applying theory of negatively dependent random variables to obtain several new concentration results. We also propose a novel extensions of the method leveraging number theory techniques and particle algorithms, called Near-Orthogonal Monte Carlo (NOMC). We show that NOMC is the first algorithm consistently outperforming OMC in applications ranging from kernel methods to approximating distances in probabilistic metric spaces.
22 pages, 4 figures
References in corpus (9)
- Orthogonal Random Features
- The Unreasonable Effectiveness of Structured Random Orthogonal Embeddings
- Recycling Randomness with Structure for Sublinear time Kernel Expansions
- Orthogonal Estimation of Wasserstein Distances
- On two ways to use determinantal point processes for Monte Carlo integration
- Rapidly Adaptable Legged Robots via Evolutionary Meta-Learning
- Introduction to Optimal Transport Theory
- Robotic Table Tennis with Model-Free Reinforcement Learning
- Structured Monte Carlo Sampling for Nonisotropic Distributions via Determinantal Point Processes